The black hole singularities are of two types:
- Spacelike--- this is only for nonrotating uncharged black hole
- Timelike--- exact solutions for everything else, rotating charged
- Cauchy horizon--- this is a makeshift fix to get rid of 2 and turn it into 1, which is introduced in an ad-hoc way by Penrose (he wants a spacelike singularity, and doesn't get it in the exact solutions). It isn't really third option.
For the spacelike singularities, they are in the future of any point in the interior, so they are not a location, but a time. They should not be thought of as an infinitely dense source of gravity, but rather an endpoint to the interior continuation, where from the string point of view, the infalling matter has totally thermalized.
For the timelike singularities, they have a diverging stress tensor, but they are not exactly ordinary singularities. They repel ordinary matter, and only light can touch them. If you shine light on them, they uncompress it, which is required for the solution to continue into the future (by the method of proof of Penroses' theorem--- the geodesics inside the horizon are all converging, and only a singularity can convert them to diverging).
This behavior has no non-relativistic analog. The picture of the singularity as a point of infinite density is only half-way accurate for a neutral black hole, which is completely non-generic as far as classical solutions go.
I must add than in my opinion, it is only the timelike singularities which are physical, the spacelike neutral Scwarzschild singularity is an artifact of high symmetry, and turns into a timelike singularity under generic perturbations. I also believe because of this that stuff goes into a black hole, and out again, after a traversal.